X-ray phase contrast imaging (XPCi) creates images of an object utilizing changes in the phase of an x-ray beam as it passes through the object. XPCi improves over problems with conventional x-ray imaging, where poor image contrast can arise from small attenuation differences. Its application can be found in medicine (e.g., breast tissue tumors) and industrial applications. Because many substances induce a phase shift in the x-rays as they pass through an object, the detection of features not discernable by conventional x-ray imaging is possible with XPCi. The phase shift can be measured using an interferometer to obtain high-contrast data for low-absorption objects.
FIG. 1 depicts standard XPCi imaging process 100. One implementation of XPCi is a grating based technique, where a series of images is acquired along with a grating stepping technique in order to sample the interference patterns (i.e., fringes) for each detector pixel (x,y). To correct for system non-homogeneities, sampling is performed for an air scan 105 to create a reference of system contributions. An object scan 110 is performed to collect data for imaging.
The fringes can be well approximated as a sinusoidal curve. This curve is analyzed by performing Fourier analysis 115. The Fourier analysis computes for the air scan and the object scan the first two (complex) Fourier coefficients a(x,y,0) and a(x,y,1) (0th and 1st order, respectively) 120, 125. In principle the analysis is not limited to the first two coefficients but can be done up to order N. An exemplary, conventional formulation for process 100 can be expressed asminρ∥Hρ−b(x,y,•)∥2a(x,y,•)=ρ  (EQ.1)
where (x, y) is pixel position;
 the Fourier transform up to order N; and
H is the adjoint of .
Since  is an orthogonal operator, the adjoint equals the inverse Fourier transform, and equation 1 can be solved directly as a(x,y,•)=b(x, y,•)
The three types of contrast provided by XPCi (i.e., absorption image att(x,y) 132, differential phase image dpc(x,y) 134, and dark field image dci(x,y) 138), can be computed by a complex division 130 of the Fourier coefficients of the object scan by the Fourier coefficients of the air scan, as follows
                                          p            ⁡                          (                              x                ,                y                ,                n                            )                                =                                                    a                obj                            ⁡                              (                                  x                  ,                  y                  ,                  n                                )                                                                    a                air                            ⁡                              (                                  x                  ,                  y                  ,                  n                                )                                                    ,                  n          =                      0            ⁢                                                  ⁢            …            ⁢                                                  ⁢            N                                              (                  EQ          .                                          ⁢          2                )                                          att          ⁡                      (                          x              ,              y                        )                          =                                        p            ⁡                          (                              x                ,                y                ,                0                            )                                                                    (                  EQ          .                                          ⁢          3                )                                          dpc          ⁡                      (                          x              ,              y                        )                          =                            ⁢                      (                          p              ⁡                              (                                  x                  ,                  y                  ,                  1                                )                                      )                                              (                  EQ          .                                          ⁢          4                )                                          dci          ⁡                      (                          x              ,              y                        )                          =                                            ∑                                                n                  =                  1                                ,                3                ,                5                ,                                                                  ⁢                …                ⁢                                                                  ,                                                                  ⁢                N                                      ⁢                                                  ⁢                                                        p                ⁡                                  (                                      x                    ,                    y                    ,                    n                                    )                                                                                                                  p              ⁡                              (                                  x                  ,                  y                  ,                  0                                )                                                                                    (                  EQ          .                                          ⁢          5                )            
Subsequently, phase image 150 can be computed by integrating 140 the differential phase image 123 on a line-by-line basis. However, integration step 140 is an optional part of the process.
Advanced processing strategies of XPCi images (e.g., regularization/de-noising using neighboring pixels/rows, etc.) have only been applied to the later steps of conventional process 100.